This question is not meant to presume that this idea is obviously a good one, or obviously desired by the residents of the area.
I notice that just west of Manhattan Island is an island-y chunk of land in Jersey. It has a few separate areas on it; Bayonne in the south, Jersey City in the lower-middle, Hoboken and Union City in the upper-middle, North Bergen higher up, and then a bunch of small communities. It's just across the river from Manhattan. The area seems to lack a subway and seems much lower density than NYC.
I'm talking specifically about all of Bergen and Hudson County.
My question is, if those areas incorporated into a city and built a transit system like NYC (or specifically Manhattan), would it quickly come to feel like just another part of or a twin of Manhattan? If so, how long would it take?